Where is it easier to solve the problem of quadratic functions?

My experience is that the relationship between images and analytic forms must be grasped first.

For example, if the quadratic term coefficient a is greater than 0, the image opening is upward.

In addition, it is necessary to find the axis of symmetry, which requires a certain amount of formula skills, and the first floor introduces the most basic formula method.

The advantage of the recipe is that the axes of symmetry and the maximum value of the function can be seen at a glance.

In addition, it is necessary to be able to flexibly set the equation according to the conditions given in the problem.

Specifically, there are general formulas, vertex formulas, intersection formulas, etc. (since I am a high school student, I only remember the commonly used ones) There may be different names for different teachers, so ask the teacher what kind of formula they should know.

One of the three major ideas of mathematics is the combination of numbers and shapes.

Drawing graphs is very helpful for solving problems and can also help you check whether your answers are right or wrong.

Step 1Propose the quadratic coefficients.

2.In parentheses, add half of the square of the primary term coefficient and subtract it to ensure that the value does not change.

3.That's when you find the perfect square. Then multiply the quadratic coefficient.

For example: y=2x -12x+7

2(x -6x+ — proposes a quadratic coefficient "2".

2(x -6x+9+ —half of the square of 6 is 9, plus 9 and subtracting 2 after it [(x-3) x -6x+9 is perfectly squared, equal to (x-3) 2(x-3) -11 - the quadratic coefficients are multiplied again, so the vertex coordinates of this quadratic function are (3,-11).

If you don't know how to solve it very well, you can send the graph first, and the vertex coordinates should be known (I don't know how to read the book), and combine the graph to solve it.

The practical problems that quadratic functions can solve are as follows:

1. Application in bridge construction, parabola has a wide range of applications in bridge construction. In real life, due to various needs, most bridge buildings use the properties of quadratic functions and design their shapes in the form of parabolas.

2. The application of quadratic functions in economic life is mainly divided into different aspects such as investment strategy, sales pricing, goods storage, consumption and accommodation, etc., and these different aspects have a common point, that is, the problem of maximizing profits. Whether it is investment or sales, profit is our primary concern.

3. Application in daily life, in addition to the application of quadratic functions in architectural design and economic life, the application of quadratic functions in daily life is also very extensive. The path of the sphere of various sports that we participate in in our daily life is a parabola. In the course of sports, the estimation of the athlete's performance and the accuracy of the ball hit is inseparable from the quadratic function.

4. The application of policy subsidies, the living subsidies for urban and rural residents in the society, the construction of urban planning, and the construction requirements of public facilities all involve the application of quadratic functions.

f(x)=ax^2+bx+c

The formula for finding the root (any one homoquadratic function is fine): δb 2-4ac, the discriminant formula of the root (if δ< 0, there is no real solution to this equation; If δ=0, there is only one solution to this equation; If δ> 0, there are 2 different solutions to this equation).

x=(-b±√δ2a

Cross multiplication: f(x)=(kx+a)(kx+b).

1. a>0, b>0, c<0

2. If the vertices of the quadratic function y=—x 2—2x+c are below the x-axis, then the value range of c is c<-1

3. The coordinates of the intersection of the hyperbola y=-4 x and the parabola y=x 2-2x-2 are (2,-2), (2,2 2) and (2,-2 2).

2。Discriminant should be <0. So c<-1.

3.(2,-2), (2,2 2) and (2,-2 2).

Wouldn't you have any basic knowledge of these quadratic functions?! ...

Substituting 1 and 2 into the function to calculate the value, the number obtained is less than 0, and the root is not in the range of (1,2).

Substituting x=3 into Eq. 0, we can see that the root is in the range of (2,3).

a(x+b 2a) 2-(b 2-4ac) 4a axis of symmetry straight line x=-b 2a

The fixed-point coordinates (-b 2a, (4ac-b 2) 4a) are separated from each other.

Tangent Intersects.

The unary quadratic equation is ax 2+bx+c=01: axis of symmetry x=0, fixed-point(0,0)2: axisof symmetry(0,3)3:

Axis of Symmetry x=1, fixed point (1,0)4: axis of symmetry x=1, fixed point (1,3)5: axis of symmetry x=2, fixed point (2,-6) quadratic function y=2x +x-3 opening direction ( up ) axis of symmetry straight line (x=-1 4 ) vertex coordinates (-1 4, -25 8 ) quadratic function y=x Move 2 units to the left and then move down by one unit, then the expression of the new function is y=(x+2) 1 its opening direction (

Up ) axis of symmetry (straight line x=-2).

Vertex coordinates (-2,-1)quadratic function y=x +3x-4 image with x-axis has ( 2

The intersection, they didn't believe was: (-4,0).

1,0) If the parabola y=2x +8x+m has only one intersection point with the x-axis, they are ( (2,0)

The parabola y=-3 4x is open to ( down.

To the left of the axis of symmetry, y increases with the ( of x.

And if the quadratic function y=2x is shifted downwards by 3 units and to the right by 4 units, the parabolic relationship is obtained: y=2(x-4) 2-3 quadratic function y=ax +bx+c, if a o, when x=( b 2a

, the function y has the smallest (.

value, whose value is.

4ac-b^2)/4a

If a 0 when x = ( b 2a

, the function y has the largest ( large.

value, which has a value of (4AC-B2)4A

The quadratic function y=x +2x+3 has the smallest (.

value when x=(1

The maximum value of the function is ( 2

The quadratic function -y=x +2x+3 has the largest (.

value when x=(1

The maximum value of the time function is ( -2

aDecide on the direction of the opening. Up when it is greater than 0, and down when it is less than 0.

b (-2a) is the abscissa of the vertex, greater than 0 on the right side of the y-axis, and less than 0 on the left side of the y-axis.

c is the ordinate of the intersection point with the y-axis, greater than 0 at the top of the x-axis, and less than 0 at the bottom of the x-axis.

A determines the direction of the opening, AB jointly determines the axis of symmetry, and C determines the intersection of the image with the y-axis.